VIEWS: 16 PAGES: 31 POSTED ON: 4/26/2011
Succinct Data Structures Ian Munro 1 May 09 General Motivation In Many Computations ... Storage Costs of Pointers and Other Structures Dominate that of Real Data Often this information is not “just random pointers” How do we encode a combinatorial object (e.g. a tree) of specialized information … even a static one in a small amount of space & still perform queries in constant time ??? 2 May 09 Succinct Data Structure Representation of a combinatorial object: Space requirement “close to” information theoretic lower bound and Time for operations required of the data type comparable to that of representation without such space constraints (O(1)) 3 May 09 Example : Static Bounded Subset Given: Universe of n elements [0,...n-1] and m arbitrary elements from this universe Create: a static structure to support search in constant time (lg n bit word & usual ops) Using: Essentially minimum possible # bits n … lg((m)) Operation: Member query in O(1) time (Brodnik & M.) 4 May 09 Careful .. Lower Bounds Beame-Fich: Find largest less than i is tough in some ranges of m (e.g. m≈2 √lg n) But OK if i is present this can be added (Raman, Raman, Rao) 5 May 09 Focus on Trees .. Because Computer Science is .. Arbophilic Directories (Unix, all the rest) Search trees (B-trees, binary search trees, digital trees or tries) Graph structures (we do a tree based search) and a key application Search indices for text (including DNA) 6 May 09 Preprocess Text for Search A Big Patricia Trie/Suffix Trie 0 1 0 1 100011 Given a large text file; treat it as bit vector Construct a trie with leaves pointing to unique locations in text that “match” path in trie (paths must start at character boundaries) Skip the nodes where there is no branching ( n-1 internal nodes) 7 May 09 Space for Trees Abstract data type: binary tree Size: n-1 internal nodes, n leaves Operations: child, parent, subtree size, leaf data Motivation: “Obvious” representation of an n node tree takes about 6 n lg n bit words (up, left, right, size, memory manager, leaf reference) i.e. full suffix tree takes about 5 or 6 times the space of suffix array (i.e. leaf references only) 8 May 09 Succinct Representations of Trees Start with Jacobson, then others: Catalan number = # ordered rooted trees, same number of binary trees ≈ 4n/(πn)3/2 Lower bound on specifying is about 2n bits What are natural representations? 9 May 09 Arbitrary Order Trees Use parenthesis notation Represent the tree As the binary string (((())())((())()())): traverse tree as “(“ for node, then subtrees, then “)” Each node takes 2 bits 10 May 09 About Heaps Only 1 heap (shape) on n nodes 1 Balanced tree, 2 3 bottom level pushed left number nodes row by row; 4 5 6 7 lchild(i)=2i; rchild(i)=2i+1 8 9 10 11 May 09 About Heaps Only 1 heap (shape) on n nodes 18 1 Balanced tree, 12 16 2 3 bottom level pushed left 6 4 number nodes row by row; 4 10 5 6 15 7 lchild(i)=2i; rchild(i)=2i+1 1 5 9 8 9 10 Data: Parent value > child This gives an implicit data structure for priority queue 12 May 09 Generalizing: Heap-like Notation for any Binary Tree Add external nodes 1 Enumerate level by level 1 1 1 0 1 1 1 0 0 0 0 1 0 0 0 0 Store vector 11110111001000000 length 2n+1 (Here we don’t know size of subtrees; can be overcome. Could use isomorphism to flip between notations) 13 May 09 How do we Navigate? Jacobson’s key suggestion: Operations on a bit vector rank(x) = # 1’s up to & including x select(x) = position of xth 1 So in the binary tree leftchild(x) = 2 rank(x) rightchild(x) = 2 rank(x) + 1 parent(x) = select(x/2) 14 May 09 Heap-like Notation for a Binary Tree Add external nodes 1 Enumerate level by level 1 1 Rank 5 Node 11 1 0 1 1 1 0 0 0 0 1 0 0 0 0 Store vector 11110111001000000 length 2n+1 (Here don’t know size of subtrees; can be overcome. Can also use isomorphism to flip between notations) 15 May 09 Rank & Select Rank: Auxiliary storage ~ 2nlglg n / lg n bits #1’s up to each (lg n)2 rd bit #1’s within these too each lg nth bit Table lookup after that Select: More complicated (especially to get this lower order term) but similar notions Key issue: Rank & Select take O(1) time with lg n bit word (M. et al) 16 May 09 Lower Bound: for Rank & for Select Theorem (Golynski): Given a bit vector of length n and an “index” (extra data) of size r bits, let t be the number of bits probed to perform rank (or select) then: r=Ω(n (lg t)/t). Proof idea: Argue to reconstructing the entire string with too few rank queries (similarly for select) Corollary (Golynski): Under the lg n bit RAM model, an index of size (n lglg n/ lg n) is necessary and sufficient to perform the rank and the select operations. 17 May 09 Other Combinatorial Objects Planar Graphs (Lu et al; Barbay et al) Permutations [n]→ [n] Or more generally Functions [n] → [n] But what operations? Clearly π(i), but also π -1(i) And then π k(i) and π -k(i) Suffix Arrays (special permutations; references to positions in text sorted lexicographically) in linear space Arbitrary Graphs (Farzan & M) 18 May 09 Permutations: a Shortcut Notation Let P be a simple array giving π; P[i] = π[i] Also have B[i] be a pointer t positions back in (the cycle of) the permutation; B[i]= π-t[i] .. But only define B for every tth position in cycle. (t is a constant; ignore cycle length “round-off”) 2 4 5 13 1 8 3 12 10 So array representation P = [8 4 12 5 13 x x 3 x 2 x 10 1] 1 2 3 4 5 6 7 8 9 10 11 12 13 19 May 09 Representing Shortcuts In a cycle there is a B every t positions … But these positions can be in arbitrary order Which i’s have a B, and how do we store it? Keep a vector of all positions: 0 = no B 1 = B Rank gives the position of B[“i”] in B array So: π(i) & π -1(i) in O(1) time & (1+ε)n lg n bits Theorem: Under a pointer machine model with space (1+ ε) n references, we need time 1/ε to answer π and π -1 queries; i.e. this is as good as it gets … in the pointer model. 20 May 09 Getting n lg n Bits This is the best we can do for O(1) operations But using Benes networks: 1-Benes network is a 2 input/2 output switch r+1-Benes network … join tops to tops #bits(n)=2#bits(n/2)+n=n lg n-n+1=min+(n) 1 3 2 5 R-Benes Network 3 7 4 8 5 1 6 6 R-Benes Network 7 4 8 2 21 May 09 A Benes Network Realizing the permutation (std π(i) notation) π = (5 8 1 7 2 6 3 4) ; π-1 = (3 5 7 8 1 6 4 2) Note: (n) bits more than “necessary” 1 3 2 5 3 7 4 8 5 1 6 6 7 4 8 2 22 May 09 What can we do with it? Divide into blocks of lg lg n gates … encode their actions in a word. Taking advantage of regularity of address mechanism and also Modify approach to avoid power of 2 issue Can trace a path in time O(lg n/(lg lg n) This is the best time we are able get for π and π-1 in nearly minimum space. 23 May 09 Both are Best Observe: This method “violates” the pointer machine lower bound by using “micropointers”. But … More general Lower Bound (Golynski): Both methods are optimal for their respective extra space constraints (Note: backpointer approach took (n lg n) extra bits) 24 May 09 Back to the main track: Powers of π Consider the cycles of π ( 2 6 8)( 3 5 9 10)( 4 1 7) Bit vector indicates start of each cycle ( 2 6 8 3 5 9 10 4 1 7) Ignore parens, view as new permutation, ψ. Note: ψ-1(i) is position containing i … So we have ψ and ψ-1 as before Use ψ-1(i) to find i, then bit vector (rank, select) to find πk or π-k 25 May 09 Functions Now consider arbitrary functions [n]→[n] “A function is just a hairy permutation” All tree edges lead to a cycle 26 May 09 Challenges here Essentially write down the components in a convenient order and use the n lg n bits to describe the mapping (as per permutations) To get fk(i): Find the level ancestor (k levels up) in a tree or Go up to root and apply f the remaining number of steps around a cycle 27 May 09 Level Ancestors There are several level ancestor techniques using O(1) time and O(n) WORDS. Adapt Bender & Farach-Colton to work in O(n) bits But going the other way … 28 May 09 Level Ancestors Moving Down the tree requires care f-3( ) = ( ) The trick: Report all nodes on a given level of a tree in time proportional to the number of nodes, and Don’t waste time on trees with no answers 29 May 09 Final Function Result Given an arbitrary function f: [n]→[n] With an n lg n + O(n) bit representation we can compute fk(i) in O(1) time and f-k(i) in time O(1 + size of answer). f & f-1 are very useful in several applications … then on to binary relations (HTML markup) 30 May 09 Conclusion Interesting, and useful, combinatorial objects can be: Stored succinctly … O(lower bound) +o() So that Natural queries are performed in O(1) time (or at least very close) Programs: http://pizzachili.dcc.uchile.cl/index.html This can make the difference between using the data type and not … 31 May 09